Non necessary to solve the question, but with the other hints, once you have solved this specific example, you might also be able to explore which numbers of zeros are impossible for factorials [when would you add more than one?]
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Mark BennetSep 13 '11 at 6:58

4 Answers
4

Partial answer: for any $k \ge 2$, $k!$ will have a higher power of $2$ than the power of $5$ in its prime factorization, so new zeros will be added exactly when $k$ is a multiple of $5$. If there is any value of $k!$ with 99 zeros at the end of its decimal representation, then there is a first one, call it $k_{99}!$, and $k_{99}$ is a multiple of $5$. $(k_{99}+j)!$ for $j=1,2,3,4$ also must have 99 zeros, but $(k_{99}+5)!$ will have more than 99 zeros, because the $k_{99}+5$ term is a multiple of $5$ and will match with at least one existing factor of 2 to create one or more new zeros at the end of the decimal representation. So there are either 0 or 5 values of $k!$ ending in 99 zeros and all that's left is to determine which case it is.

For extra credit, see if there is a way to find the answer that is easier than actually trying to compute the value of $k_{99}$.

Hint. A number ends in $0$ if and only if it is a multiple of $10$. A number ends in two zeroes if and only if it is a multiple of $100=2^2\times 5^2$. A number ends in three zeroes if and only if it is a multiple of $1000 = 2^3\times 5^3$. Etc. So, e.g., a number ends in exactly two zeroes if and only if it is a multiple of $100$ but not of $1000$.

In order that k! end in 99 zeroes, it must have 99 factors of 10 and so 99 2's and 99 5's. Since there are going to be fewer 5's than 2's, k! must have exactly 99 5's. We first get that for k= 99*5= 495 and then get 100 5's for k= 100*5= 500. Any number from 495 to 499 will have exactly 99 5's in its factorial. There are 5 such numbers.

Actually, the answer is correct but the process has a little problem. 25=5^2 provides double 5. 125=5^3 provides triple 5. However,100 and 99 minus 3(1+2 extra 5) at the same time. Hence, it has the same answer with the solution above.